U.S.-U.K. Cooperative Research: Applications of Brownian Motion to Differential Geometry

这项长期访问英国的奖学金将支持波士顿大学的Steven Rosenberg、华威大学的K.D. Elworthy和苏格兰爱丁堡大学的T. Lyons在几何分析和微分几何方面的合作研究。研究者提出将概率和经典几何方法应用于微分几何问题。他们将学习黎曼流形及其在流形拓扑和几何中的应用。本文将研究三个问题：(1)负弯曲流形欧拉特性的Hopf猜想；(2)推广正弯曲流形Bochner和Myers定理的一种方法；(3)一种利用Fukaya方法的修正来产生拉普拉斯算子的软特征值估计的方法。后面的问题将由爱丁堡大学的T. Lyons博士来解决。流形是数学和物理许多领域的基本概念。通过结合美国和英国研究者在经典微分几何和概率方面的互补专业知识，这种合作提供了一种创新的方法来研究这些问题。

This award for a long term visit to the United Kingdom will support collaborative research in geometric analysis and differential geometry between Steven Rosenberg of Boston University, K.D. Elworthy of the University of Warwick, and T. Lyons of the University of Edinburgh, Scotland. The investigators propose to apply probabilistic and classical geometric methods to problems in differential geometry. They will study Riemannian manifolds and their applications to topology and geometry of manifolds. Three problems will be investigated: (1) The Hopf conjecture for the Euler characteristic of negatively curved manifolds; (2) a method for extending Bochner's and Myers' theorems for positively curved manifolds; and (3) a method for yeilding soft eigenvalue estimates for the Laplacian using a modification of the Fukaya method. The later problem will be developed with Dr. T. Lyons of the University of Edinburgh. Manifolds are a fundamental concept in many areas of mathematics and physics. This collaboration offers an innovative approach to studying these problems by combining the complementary expertise of the U.S. and British investigators in classical differential geometry and in probability respectively.